耶鲁大学政治系《高级建模理论：博弈论》讲义.doc

耶鲁大学政治系《高级建模理论：博弈论》讲义 1StrategicFormGameStrategicformgames,sometimescallednormalformgames,aretypicallyusedtomodelstaticinteractionsofplayers.Staticenvironmentsarethoseinwhichthetimingofplayers’choicesdonotmatter.Wemaythinkofthemsimplyassituationsinwhichallplayersmovesimultaneously.1.1ElementsofGame??Setofplayers:N.–notationsforagenericelementofN:i,j,k,etc.–Wewillmostlyassumethatthereareafinitenumberofplayers.Inthatcase,N={1,...,n}.??Setofavailablestrategiesforeachplayeri:Si.–notationsforagenericelementofS′′i:si,s′,s,s?a name=baidusnap8#0046990?iii,etc.–aprofileofstrategies:s=(s1,...,sn).Moregenerally,s=(si)i∈N.–Setofallprofilesofstrategies:S=S1×···×Sn.Moregenerally,S=×i∈NSi.??Payofffunctionforeachplayer:ui:S→R.–ui(s1,...,sn):thepayoffplayeriwouldreceiveifplayer1choosess1,player2choosess2,...,playernchoosessn.Noticethatthedomainofeachperson’spayofffunctionisthesetofstrategyprofiles,notthesetofindividualstrategies.Inotherwords,howmuchIgetfromagamedependsnotjustonmystrategybutalsoonmyopponents’strategies.–Avector-valuedpayofffunctionu:S→Rnthatcollectsallplayers’payofffunctions.Foralls∈S,u(s)=(u1(s),...,un(s)).AstrategicformgameisatripleG=(N,S,u).AgameGisafinitegameifbothNandSarefinitesets.Forawhile,wewillfocusonfinitegames.1 Example1Rock,Paper,ScissorsPlayer1andplayer2simultaneouslychooseeitherRock(R),Paper(P),orScissors(C).RbeatsC,CbeatsP,andPbeatsR.Eachplayergetspayoffoneifshewins,negativeoneifsheloses,zeroiftheoutcomeisatie.Asweapplythenotationsdefinedabove:N={1,2},S1=S2={R,P,C},S={(R,R),(R,P),(R,C),(P,R),(P,P),(P,C),(C,R),(C,P),(C,C)}u(R,C)=u(P,R)=u(C,P)=(1,??1),u(R,R)=u(P,P)=u(C,C)=(#0046990,#0046990),u(R,P)=u(P,C)=u(C,R)=(??1,1).Conveniently,thisgameisrepresentedbythefollowinggamematrix.Player2RPCR#0046990,#0046990??1,11,??1Player1P1,??1#0046990,#0046990??1,1C??1,11,??1#0046990,#0046990AdditionalNotationWeoftenwilldiscussvaryingasingleplayer’sstrategywhileholdingtheotherplayers’strategiesfixed.Todothis,weusenotations??itodenoteaprofileofstrategeisofallplayersexce